Abstract
A Shilla graph is defined as a distance-regular graph of diameter 3 with second eigen-value θ1 equal to a3. For a Shilla graph, let us put a = a3 and b = k/a. It is proved in this paper that a Shilla graph with b2 = c2 and noninteger eigenvalues has the following intersection array:
If Γ is a Q-polynomial Shilla graph with b2 = c2 and b = 2r, then the graph Γ has intersection array
and, for any vertex u in Γ, the subgraph Γ3(u) is an antipodal distance-regular graph with intersection array
The Shilla graphs with b2 = c2 and b = 4 are also classified in the paper.
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References
J. H. Koolen and J. Park, “Shilla distance-regular graphs,” European J. Combin. 31 (8), 2064–2073 (2010).
A. A. Makhnev and D. V. Paduchikh, “On the automorphism group of the distance-regular graph with intersection array {24, 21, 3; 1, 3, 18},” Algebra i Logika, 51 (4), 476–495 (2012). [Algebra and Logic 51 (4), 319–332 (2012)].
A. Jurišićand J. Vidali, “Extremal 1-codes in distance-regular graphs of diameter 3,” Des. Codes Cryptogr. 65 (1-2), 29–47 (2012).
J. Vidali, Kode v razdaljno regularnih grafih, Doctoral Dissertation (Univerza v Ljubljani, Ljubljana, 2013).
A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-Regular Graphs (Springer-Verlag, Berlin, 1989).
J. H. Koolen, J. Park, and H. Yu, “An inequality involving the second largest and smallest eigenvalue of a distance-regular graph,” Linear Algebra Appl. 434 (12), 2404–2412 (2011).
J. H. Koolen and J. Park, “Distance-regular graphs with large a1 or c2 at least half the valency,” J. Combin. Theory Ser. A 119 (3), 546–555 (2012).
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Original Russian Text © A. A. Makhnev, M. S. Nirova, 2018, published in Matematicheskie Zametki, 2018, Vol. 103, No. 5, pp. 730–744.
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Makhnev, A.A., Nirova, M.S. Distance-Regular Shilla Graphs with b2 = c2. Math Notes 103, 780–792 (2018). https://doi.org/10.1134/S0001434618050103
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DOI: https://doi.org/10.1134/S0001434618050103